Explanation

(c)
Let $a_{n} = \frac{1}{(n + 1)(n + 2)\ldots(n + k)}$
$\Rightarrow a_{n} = \frac{1}{(k - 1)}\left( \frac{(n + k) - (n + 1)}{(n + 1)(n + 2)\ldots(n + k)} \right)$
$\Rightarrow \ \frac{1}{(k - 1)}\left( \frac{1}{(n + 1)(n + 2)\ldots(n + k - 1)} \right) - \left( \frac{1}{(n + 2)(n + 3)\ldots(n + k)} \right)$
$\therefore\ S_{n} = a_{1} + a_{2} + ... + a_{n}$
$= \frac{1}{(k - 1)}\left( \frac{1}{2 \bullet 3 \bullet 4\ldots k} - \frac{1}{(n + 2)(n + 3)\ldots(n + k)} \right)$
$\Rightarrow \ \ \lim_{n \rightarrow \infty}{S_{n} =}\frac{1}{(k - 1)k!}$